2 problems

Let $A$ be an $R$-algebra. Write $F_{R\text{-alg}}(A,A)$ for the monoid of $R$-algebra self-maps, $B F_{R\text{-alg}}(A,A)$ for its classifying space, and ${\bf Der}_R(A,A)$ for th…

Let $R$ be a commutative ring spectrum, let $A$ and $B$ be $R$-algebras, and let $f:A\rightarrow B$ be an $R$-algebra map. Write $F_{R\text{-alg}}(A,B)$ for the space of $R$-algebr…